Question:The linear function g is defined by \(\mathrm{g(x) = 4 - 7x}\). What is the y-coordinate of the y-intercept of...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = 4 - 7x}\)
  • We need the y-coordinate of the y-intercept of \(\mathrm{y = 3g(x) + 2}\)

• What this tells us: The y-intercept occurs when \(\mathrm{x = 0}\), so we need to find the value of y when \(\mathrm{x = 0}\).

2. INFER the solution strategy

• To find y when \(\mathrm{x = 0}\), we need to substitute \(\mathrm{x = 0}\) into \(\mathrm{y = 3g(x) + 2}\)
• But first, we must evaluate \(\mathrm{g(0)}\) before we can complete the substitution
• This means we work from the inside out: find \(\mathrm{g(0)}\), then use that result

3. SIMPLIFY through the calculations

• First, find \(\mathrm{g(0)}\):
  \(\mathrm{g(0) = 4 - 7(0)}\)
  \(\mathrm{= 4 - 0}\)
  \(\mathrm{= 4}\)

• Now substitute \(\mathrm{g(0) = 4}\) into the main equation:
  \(\mathrm{y = 3g(0) + 2}\)
  \(\mathrm{= 3(4) + 2}\)
  \(\mathrm{= 12 + 2}\)
  \(\mathrm{= 14}\)

Answer: 14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "y-intercept" means "substitute \(\mathrm{x = 0}\)." Instead, they might try to set the entire expression equal to zero or look for where the function crosses the x-axis.

This leads to confusion and guessing, as they don't have a clear starting point for the solution.

Second Most Common Error:

Poor INFER reasoning: Students might recognize they need \(\mathrm{x = 0}\), but try to substitute directly into the full expression \(\mathrm{y = 3g(x) + 2}\) without first evaluating \(\mathrm{g(0)}\). They might write something like \(\mathrm{y = 3g(0) + 2}\) and then get stuck because they haven't computed what \(\mathrm{g(0)}\) actually equals.

This causes them to abandon systematic solution and guess.

The Bottom Line:

This problem tests whether students understand the fundamental concept of intercepts AND can handle function composition systematically. The key insight is recognizing that you must work from the inside out—evaluate the inner function first, then use that result in the outer expression.